On the EIT problem for nonorientable surfaces

Abstract

Let $(\Omega ,g)$ be a smooth compact two-dimensional Riemannian manifold with boundary, ${\Lambda }_g:f\mapsto {\partial }_{\nu }u{|}_{\partial \Omega }$ its DN map, where $u$ obeys ${\Delta }_{g}u=0$ in $\Omega$ and $u{|}_{\partial \Omega }=f$. The Electric Impedance Tomography problem is to determine $\Omega$ from ${\Lambda }_g$. A criterion is proposed that enables one to detect (via ${\Lambda }_g$) whether $\Omega$ is orientable or not. The algebraic version of the BC-method is applied to solve the EIT problem for the Moebius band. The main instrument is the algebra of holomorphic functions on the double covering $\mathbb{M}$ of $M$, which is determined by ${\Lambda }_g$ up to an isometric isomorphism. Its Gelfand spectrum (the set of characters) plays the role of the material for constructing a relevant copy $(M^{\prime },g^{\prime })$ of $(M,g)$. This copy is conformally equivalent to the original, provides $\partial M^{\prime }=\partial M,{\Lambda }_{g^{\prime }}={\Lambda }_g$, and thus solves the problem.

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References in corpus (1)

• On algebraic and uniqueness properties of 3d harmonic quaternion fields